An adaptive response surface method for continuous Bayesian model calibration

Abstract

Non-linear numerical models of the injection phase of a carbon sequestration (CS) project are computationally demanding. Thus, the computational cost of the calibration of these models using sampling-based solutions can be formidable. The Bayesian adaptive response surface method (BARSM)—an adaptive response surface method (RSM)—is developed to mitigate the cost of sampling-based, continuous calibration of CS models. It is demonstrated that the adaptive scheme has a negligible effect on accuracy, while providing a significant increase in efficiency. In the BARSM, a meta-model replaces the computationally costly full model during the majority of the calibration cycles. In the remaining cycles, the full model is used and samples of these cycles are utilized for adaptively updating the meta-model. The idea behind the BARSM is to take advantage of the fact that sampling-based calibration algorithms typically tend to sample more frequently from areas with a larger posterior density than from areas with a smaller posterior density. This behavior of the sampling-based calibration algorithms is used to adaptively update the meta-model and to make it more accurate where it is most likely to be evaluated. The BARSM is integrated with Unscented Importance Sampling (UIS) (Sarkarfarshi and Gracie, Stoch Env Res Risk Assess 29: 975–993, 2015), which is an efficient Bayesian calibration algorithm. A synthesized case of supercritical CO2 injection in a heterogeneous saline aquifer is used to assess the performance of the BARSM and to compare it with a classical non-adaptive RSM approach and Bayesian calibration method UIS without using RSM. The BARSM is shown to reduce the computational cost compared to non-adaptive Bayesian calibration by 87 %, with negligible effect on accuracy. It is demonstrated that the error of the meta-model fitted using the BARSM, when samples are drawn from the posterior parameter distribution, is negligible and smaller than the monitoring error.